Traditionally, in Magnetic Data Storage Channels, separate Digital Data Codes control the written Recorded Magnetization Patterns and its read Data Integrity, they are respectively labeled as “Modulation” and “Error Correction” Codes. Nonlinear Modulation Codes, efficiently implemented by Finite State Automata, with 30% redundancy were used throughout the Magnetic Data Storage Industry through the early 90's. However, due to the fact that the redundancy of these Nonlinear Modulation Codes could never be effectively used for algebraic error correction, current Modulation Codes redundancy is less than 1%. On the other hand, due to their practical disappearance, coding gains are severely limited by Nonlinear Magnetization Noise, which they were originally designed to minimize. In this talk I describe a class of Integrable Dynamical Systems, whose Dynamical Trajectories, effectively generated by Finite State Automata, can be labeled by Finite Field Codewords of a Linear Error Correcting Code. This State Space description, permits the simultaneous control of both the written Recorded Magnetization Patterns, as well as the read Data Integrity, within the same Nonlinear Modulation Code. The coding gain implication of such systems will be discussed.